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The second sentence literally means NOT(WANT(P))"I lack the desire to do anything", but by implication it usually also means WANT(NOT(P))"I want to do nothing". Since this is just an implication, it's possible for it to be cancelled in certain contexts.

I was wondering if something similar is true of 〜したくない. Take a look at this sentence:

何もしたくない。

I think this usually means WANT(NOT(P))"I want to do nothing". But is it possible for it to mean NOT(WANT(P))"I lack the desire to do anything" as well?

Supplement: A is definitely a negative volition, while B is more neutral.
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istrasciNov 14 '13 at 15:29

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I think this distinction works very well with ほしい: 何もしないでほしい vs. 何もしてほしくない. But I think that 何もしたくない would in fact more likely be "I lack the desire to do anything".
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EarthliŋNov 14 '13 at 16:04

I think that negating the object (I want to not do it / それをしないことしたい) and negating the helping verb (I don't want to do it / それをしたくない) have clearly different meanings. Negating the object is just much less common. Personally, I can't explain the differences between "I want nothing. / I don't want anything / I don't something" even to myself.
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ramsayNov 14 '13 at 16:09

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@Earthling: I do not agree with the statement. 何もしてほしくない usually, if not always, means ∀x. WANT(you do not do x), that is, the same thing as 何もしないでほしい.
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Tsuyoshi ItoNov 15 '13 at 2:26

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@Earthling: Personally I do not feel that しないでほしい is stronger than してほしくない. (If anything, I even feel that してほしくない is a little stronger for some reason.) I am not sure about the distinction between meaning and interpretation here. One may argue that logically, the literal meaning of してほしくない should be NOT(WANT(you do x)), but if so, I cannot come up with a case where it is used in this literal meaning. So while I understand your logic, I am afraid that that logic does not match the actual usage.
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Tsuyoshi ItoNov 15 '13 at 10:12

2 Answers
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The negative form …したくない means “WANT(NOT(…)).” For example, 山に登りたくない means the speaker wants to avoid climbing a mountain.

To express “NOT(WANT(…)),” we have to use other constructs such as 山に登りたいのではない.

Compare the following examples.

竜とは戦いたくないが、姫を助ける方法はほかにない。

竜と戦いたいわけではないが、姫を助ける方法はほかにない。

In the first example, the speaker wants to avoid fighting against the dragon. In the second example, the speaker is just saying that it is not that he/she wants to fight against the dragon.

Your notation related to P = “do anything” is ambiguous because it does not specify how the negation interacts with the quantifier implicit in word anything. To avoid this, we should make the quantifier explicit. Then 何もしたくない means ∀x. WANT(NOT(do x)), that is, the speaker wants to be free from all actions.

Yes, it can have either meaning. English allows a distinction due to the fact that there are two clauses involved (removing the negative, 'I want' and 'to do (some/any)thing'), so it allows negation in either clause. Japanese has a single clause, and so negation has to end up in a single location. So the distinction is determined purely contextually - 何もしたくない can be either 'I want to do nothing' or 'I don't want to do anything'. (And if you think about it, these are extremely similar, and it's more of a quirk of English grammar that you can somehow distinguish the two in English.)

If you wanted to specify the second, you might reword it as したいことがない, 'there isn't anything I want to do'.